12 x 12 Magic Squares

There are several interesting techniques to make 12x12 magic square. The technique described on this page makes a Pan-Magic square; a set of six magic carpets are combined.

0	0	1	1	0	0	1	1	0	0	1	1
1	1	0	0	1	1	0	0	1	1	0	0
0	0	1	1	0	0	1	1	0	0	1	1
1	1	0	0	1	1	0	0	1	1	0	0
0	0	1	1	0	0	1	1	0	0	1	1
1	1	0	0	1	1	0	0	1	1	0	0
0	0	1	1	0	0	1	1	0	0	1	1
1	1	0	0	1	1	0	0	1	1	0	0
0	0	1	1	0	0	1	1	0	0	1	1
1	1	0	0	1	1	0	0	1	1	0	0
0	0	1	1	0	0	1	1	0	0	1	1
1	1	0	0	1	1	0	0	1	1	0	0

0	1	1	0	0	1	1	0	0	1	1	0
1	0	0	1	1	0	0	1	1	0	0	1
0	1	1	0	0	1	1	0	0	1	1	0
1	0	0	1	1	0	0	1	1	0	0	1
0	1	1	0	0	1	1	0	0	1	1	0
1	0	0	1	1	0	0	1	1	0	0	1
0	1	1	0	0	1	1	0	0	1	1	0
1	0	0	1	1	0	0	1	1	0	0	1
0	1	1	0	0	1	1	0	0	1	1	0
1	0	0	1	1	0	0	1	1	0	0	1
0	1	1	0	0	1	1	0	0	1	1	0
1	0	0	1	1	0	0	1	1	0	0	1

0	2	0	2	1	1	1	1	2	0	2	0
2	0	2	0	1	1	1	1	0	2	0	2
0	2	0	2	1	1	1	1	2	0	2	0
2	0	2	0	1	1	1	1	0	2	0	2
0	2	0	2	1	1	1	1	2	0	2	0
2	0	2	0	1	1	1	1	0	2	0	2
0	2	0	2	1	1	1	1	2	0	2	0
2	0	2	0	1	1	1	1	0	2	0	2
0	2	0	2	1	1	1	1	2	0	2	0
2	0	2	0	1	1	1	1	0	2	0	2
0	2	0	2	1	1	1	1	2	0	2	0
2	0	2	0	1	1	1	1	0	2	0	2

0	1	0	1	0	1	0	1	0	1	0	1
0	1	0	1	0	1	0	1	0	1	0	1
1	0	1	0	1	0	1	0	1	0	1	0
1	0	1	0	1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1	0	1	0	1
0	1	0	1	0	1	0	1	0	1	0	1
1	0	1	0	1	0	1	0	1	0	1	0
1	0	1	0	1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1	0	1	0	1
0	1	0	1	0	1	0	1	0	1	0	1
1	0	1	0	1	0	1	0	1	0	1	0
1	0	1	0	1	0	1	0	1	0	1	0

0	1	0	1	0	1	0	1	0	1	0	1
1	0	1	0	1	0	1	0	1	0	1	0
1	0	1	0	1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1	0	1	0	1
0	1	0	1	0	1	0	1	0	1	0	1
1	0	1	0	1	0	1	0	1	0	1	0
1	0	1	0	1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1	0	1	0	1
0	1	0	1	0	1	0	1	0	1	0	1
1	0	1	0	1	0	1	0	1	0	1	0
1	0	1	0	1	0	1	0	1	0	1	0
0	1	0	1	0	1	0	1	0	1	0	1

0	2	0	2	0	2	0	2	0	2	0	2
2	0	2	0	2	0	2	0	2	0	2	0
0	2	0	2	0	2	0	2	0	2	0	2
2	0	2	0	2	0	2	0	2	0	2	0
1	1	1	1	1	1	1	1	1	1	1	1
1	1	1	1	1	1	1	1	1	1	1	1
1	1	1	1	1	1	1	1	1	1	1	1
1	1	1	1	1	1	1	1	1	1	1	1
2	0	2	0	2	0	2	0	2	0	2	0
0	2	0	2	0	2	0	2	0	2	0	2
2	0	2	0	2	0	2	0	2	0	2	0
0	2	0	2	0	2	0	2	0	2	0	2

Creating the Pan-Magic Squares

The six magic carpets above are composed of three in the top row, each of which has been rotated to provide an orthogonal version in the second row. There are other magic carpets for order 12 but the squares above are the simplest and show how a series of order 12 pan-magic squares can be formed.

Multipliers

The Table on the right shows some examples of a list of multipliers. The numbers in each row multilpy the cells in the appropriate square. The results are summed to produce the squares. The Pan-Magic squares below were obtained using the first two rows.

0	71	108	107	12	59	120	95	24	47	132	83
137	78	29	42	125	90	17	54	113	102	5	66
9	62	117	98	21	50	129	86	33	38	141	74
140	75	32	39	128	87	20	51	116	99	8	63
1	70	109	106	13	58	121	94	25	46	133	82
136	79	28	43	124	91	16	55	112	103	4	67
10	61	118	97	22	49	130	85	34	37	142	73
139	76	31	40	127	88	19	52	115	100	7	64
2	69	110	105	14	57	122	93	26	45	134	81
135	80	27	44	123	92	15	56	111	104	3	68
11	60	119	96	23	48	131	84	35	36	143	72
138	77	30	41	126	89	18	53	114	101	6	65

0	142	3	141	4	138	7	137	8	134	11	133
131	13	128	14	127	17	124	18	123	21	120	22
36	106	39	105	40	102	43	101	44	98	47	97
119	25	116	26	115	29	112	30	111	33	108	34
48	94	51	93	52	90	55	89	56	86	59	85
83	61	80	62	79	65	76	66	75	69	72	70
84	58	87	57	88	54	91	53	92	50	95	49
71	73	68	74	67	77	64	78	63	81	60	82
96	46	99	45	100	42	103	41	104	38	107	37
35	109	32	110	31	113	28	114	27	117	24	118
132	10	135	9	136	6	139	5	140	2	143	1
23	121	20	122	19	125	16	126	15	129	12	130

The 12x12 Magic Squares

Technique 1: The Magic Carpet Approach

Creating the Pan-Magic Squares

Multipliers